Fifth Property of the Euclidean Metric
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- | For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined | + | [[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]] |
+ | For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined | ||
- | <math> | + | <math>d_{ij}=||x_i-x_j||^2 |
- | + | =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ | |
- | + | =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] | |
- | + | \left[\begin{array}{cc}x_i\\x_j\end{array}\right]</math> | |
- | \left[ | + | |
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- | + | Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]] | |
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- | + | namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math> | |
+ | * <math>\sqrt{d_{ij}}\geq0\,,~~i\not= j</math> '''('''nonnegativity''')''' | ||
+ | * <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math> '''('''self-distance''')''' | ||
+ | * <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> '''('''symmetry''')''' | ||
+ | * <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\not=\!j\!\not=\!k</math> '''('''triangle inequality''')''' | ||
- | ==Fifth property of the Euclidean metric == | ||
- | '''('''Relative-angle inequality.''')''' | ||
- | Augmenting the four fundamental | + | ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== |
- | for all <math>i_{},j_{},\ell\ | + | Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>, |
- | <math>i\!<\!j\!<\!\ell | + | for all <math>i_{},j_{},\ell\not= k_{}\!\in\!\{1\ldots_{}N\}</math> , |
- | the inequalities | + | <math>i\!<\!j\!<\!\ell</math> , and for <math>N\!\geq_{\!}4</math> distinct points <math>\,\{x_k\}\,</math> , the inequalities |
<math>\begin{array}{cc} | <math>\begin{array}{cc} | ||
- | |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj | + | |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\\ |
- | \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi | + | \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\\ |
- | 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi | + | 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi |
\end{array}</math> | \end{array}</math> | ||
- | where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> is the angle between vectors at vertex <math>x_k</math> must be satisfied at each point <math>x_k</math> regardless of affine dimension. | + | where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> is the angle between vectors at vertex <math>\,x_k\,</math> , must be satisfied at each point <math>\,x_k\,</math> regardless of affine dimension. |
== References == | == References == | ||
- | * | + | * Dattorro, [http://www.meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], Meboo, 2005 |
Current revision
For a list of points in Euclidean vector space, distance-square between points and is defined
Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [Dattorro, ch.5.2]
namely, for Euclidean metric in
- (nonnegativity)
- (self-distance)
- (symmetry)
- (triangle inequality)
Fifth property of the Euclidean metric (relative-angle inequality)
Augmenting the four fundamental Euclidean metric properties in , for all , , and for distinct points , the inequalities
where is the angle between vectors at vertex , must be satisfied at each point regardless of affine dimension.
References
- Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo, 2005